Finite Group Databasebeta

← C2×C3×#997 #10596721

Subgroups — C2×C3×#997 #10596721

Order 576.

438 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#997C211yesmaximal, characteristic, direct product factor
2C2×C3×H(5,𝔽2)C311yesmaximal, characteristic, complemented
3C2×#997C313yesmaximal, direct product factor
4C22×C3×A4—43nomaximal
5C2×C3×SL(2,𝔽3)—42nomaximal
6C3×H(5,𝔽2)—32no—
7C3×H(5,𝔽2)—33no—
8C3×H(5,𝔽2)C2×C311yescharacteristic, complemented
9C2×C3×#41—32no—
10C22×C3×D4—33no—
11#997C2×C313yesdirect product factor
12C2×C3×A4—46no—
13C2×C3×A4—43no—
14C3×SL(2,𝔽3)—42no—
15C2×H(5,𝔽2)C3211yescharacteristic, complemented, Sylow
16C24×C3—31no—
17C24×C3A413yescomplemented
18C22×A4—49no—
19C3×#41—36no—
20C3×#41—36no—
21C3×#41—32no—
22C3×#41—32no—
23C2×C3×Q8A412yescomplemented
24C2×C3×D4—36no—
25C2×C3×D4—33no—
26C2×C3×D4—312no—
27C2×C3×D4—33no—
28C2×C3×D4—33no—
29C2×C3×D4—33no—
30C22×C3×C4—33no—
31C2×SL(2,𝔽3)—46no—
32C22×C32—161no—
33C3×A4—43no—
34H(5,𝔽2)—32no—
35H(5,𝔽2)—33no—
36H(5,𝔽2)C2×C3211yescharacteristic, complemented
37C2×#41—32no—
38C22×D4—33no—
39C23×C3—33no—
40C23×C3—31no—
41C23×C3—33no—
42C23×C3—33no—
43C23×C3—46no—
44C23×C3—122no—
45C23×C3C2×A413yescomplemented
46C2×A4—418no—
47C2×A4—49no—
48C3×Q8—32no—
49C3×Q8C2×A412yescomplemented
50C3×D4—312no—
51C3×D4—36no—
52C3×D4—36no—
53C2×C3×C4—33no—
54C2×C3×C4—36no—
55C2×C3×C4—33no—
56C2×C3×C4—32no—
57SL(2,𝔽3)—46no—
58C2×C32—162no—
59C2×C32—161no—
60C24—31no—
61C24C3×A413yescomplemented
62#41—36no—
63#41—36no—
64#41—32no—
65#41—32no—
66C2×Q8C3×A412yescomplemented
67C2×D4—36no—
68C2×D4—33no—
69C2×D4—312no—
70C2×D4—33no—
71C2×D4—33no—
72C2×D4—33no—
73C22×C4—33no—
74C22×C3—33no—
75C22×C3—33no—
76C22×C3—43no—
77C22×C3—66no—
78C22×C3—123no—
79C22×C3—121no—
80C22×C3—123no—
81C22×C3—163no—
82C22×C3C22⋊A411yescharacteristic
83A4—49no—
84C3×C4—32no—
85C3×C4—32no—
86C32—161noSylow
87C23—31no—
88C23—33no—
89C23—33no—
90C23—33no—
91C23—46no—
92C23—122no—
93C23C2×C3×A413yescomplemented
94Q8—32no—
95Q8C2×C3×A412yescomplemented
96D4—36no—
97D4—36no—
98D4—312no—
99C2×C4—33no—
100C2×C4—36no—
101C2×C4—33no—
102C2×C4—32no—
103C2×C3—63no—
104C2×C3—63no—
105C2×C3—166no—
106C2×C3—163no—
107C2×C3#99712yesdirect product factor
108C2×C3C2×C22⋊A411yescharacteristic
109C22—33no—
110C22—33no—
111C22—43no—
112C22—66no—
113C22—123no—
114C22—121no—
115C22—123no—
116C22C3×C22⋊A411yescharacteristic
117C4—32no—
118C4—32no—
119C3—163no—
120C3C2×#99711yescharacteristic, direct product factor
121C2—63no—
122C2—63no—
123C2C3×#99712yesdirect product factor
124C2C2×C3×C22⋊A411yescharacteristic
subgroupG/H
center, socle, hypercenterC22×C3C22⋊A4
commutator, cosocle, hypocenterH(5,𝔽2)C2×C32
FrattiniC2C2×C3×C22⋊A4
FittingC2×C3×H(5,𝔽2)C3
LCSG ▷ H(5,𝔽2)
UCSC1 ◁ C22×C3

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#997C2direct (N × H)
2. C2×C3×H(5,𝔽2)C3semidirect
3. C2×#997C3direct (N × H)
8. C3×H(5,𝔽2)C2×C3semidirect
11. #997C2×C3direct (N × H)
15. C2×H(5,𝔽2)C32semidirect
17. C24×C3A4semidirect
23. C2×C3×Q8A4semidirect
36. H(5,𝔽2)C2×C32semidirect
45. C23×C3C2×A4semidirect
49. C3×Q8C2×A4semidirect
61. C24C3×A4semidirect
66. C2×Q8C3×A4semidirect
93. C23C2×C3×A4semidirect
95. Q8C2×C3×A4semidirect
107. C2×C3#997direct (N × H)
120. C3C2×#997direct (N × H)
123. C2C3×#997direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#9972
C2×C3×H(5,𝔽2)3
C2×#9973
C22×C3×A44
C2×C3×SL(2,𝔽3)4